zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Regularization and distributional derivatives of (x 1 2 +x 2 2 ++X p 2 ) -n in p . (English) Zbl 0578.46034

Summary: Our main aim is to present the value of the distributional derivative

¯ N x 1 k 1 x 2 k 2 ···x p k p ( 1r n ),

where r=(x 1 2 +x 2 2 +···+x p 2 ) 1/2 in p , N=k 1 +k 2 +···+k p , and p,n,k 1 ,k 2 ,···,k p are positive integers. For this purpose, we first define a regularization of 1/x n in 1 , which in turn helps us to define the regularization of 1/r n in p . These regularizations are achieved as asymptotic limits of the truncated functions H(x-ϵ)/x n and H(r-ϵ)/r n as ϵ 0, plus certain terms concentrated at the origin, where H is the Heaviside function. In the process of the derivation of the distributional derivative formula mentioned, we also derive many other interesting results and introduce some simplifying notation.

46F10Operations with distributions (generalized functions)